# Lester's theorem

The Fermat points ${\displaystyle X_{13},X_{14}}$, the center ${\displaystyle X_{5}}$ of the nine-point circle (light blue), and the circumcenter ${\displaystyle X_{3}}$ of the green triangle lie on the Lester circle (black).

In Euclidean plane geometry, Lester's theorem, named after June Lester, states that in any scalene triangle, the two Fermat points, the nine-point center, and the circumcenter lie on the same circle.

## Proofs

### Gibert's proof using the Kiepert hyperbola

Lester's circle theorem follows from a more general result by B. Gibert (2000); namely, that every circle whose diameter is a chord of the Kiepert hyperbola of the triangle and is perpendicular to its Euler line passes through the Fermat points.[1] [2]

### Dao's lemma on the rectangular hyperbola

Dao's lemma on a rectangular hyperbola

In 2014, Dao Thanh Oai showed that Gibert's result follows from a property of rectangular hyperbolas. Namely, let ${\displaystyle H}$ and ${\displaystyle G}$ lie on one branch of a rectangular hyperbola ${\displaystyle S}$, and ${\displaystyle F_{+}}$ and ${\displaystyle F_{-}}$ be the two points on ${\displaystyle S}$, symmetrical about its center (antipodal points), where the tangents at ${\displaystyle S}$ are parallel to the line ${\displaystyle HG}$,

Let ${\displaystyle K_{+}}$ and ${\displaystyle K_{-}}$ two points on the hyperbola the tangents at which intersect at a point ${\displaystyle E}$ on the line ${\displaystyle HG}$. If the line ${\displaystyle K_{+}K_{-}}$ intersects ${\displaystyle HG}$ at ${\displaystyle D}$, and the perpendicular bisector of ${\displaystyle DE}$ intersects the hyperbola at ${\displaystyle G_{+}}$ and ${\displaystyle G_{-}}$, then the six points ${\displaystyle F_{+},F_{-},E,F,G_{+},G_{-}}$ lie on a circle.

To get Lester's theorem from this result, take ${\displaystyle S}$ as the Kiepert hyperbola of the triangle, take ${\displaystyle F_{+},F_{-}}$ to be its Fermat points, ${\displaystyle K_{+},K_{-}}$ be the inner and outer Vecten points, ${\displaystyle H,G}$ be the orthocenter and the centroid of the triangle.[3]

## Generalisation

A generalization Lester circle associated with Neuberg cubic: ${\displaystyle P,Q(P),X_{13},X_{14}}$ lie on a circle

A conjectured generalization of the Lester theorem was published in Encyclopedia of Triangle Centers as follows: Let ${\displaystyle P}$ be a point on the Neuberg cubic. Let ${\displaystyle P_{A}}$ be the reflection of ${\displaystyle P}$ in line ${\displaystyle BC}$, and define ${\displaystyle P_{B}}$ and ${\displaystyle P_{C}}$ cyclically. It is known that the lines ${\displaystyle AP_{A}}$, ${\displaystyle BP_{B}}$, ${\displaystyle CP_{C}}$ are concurrent. Let ${\displaystyle Q(P)}$ be the point of concurrency. Then the following 4 points lie on a circle: ${\displaystyle X_{13}}$, ${\displaystyle X_{14}}$, ${\displaystyle P}$, ${\displaystyle Q(P)}$. [4] When ${\displaystyle P=X(3)}$, it is well-known that ${\displaystyle Q(P)=Q(X_{3})=X_{5}}$, the conjecture becomes Lester theorem.

## Notes

1. ^ Paul Yiu (2010), The circles of Lester, Evans, Parry, and their generalizations. Forum Geometricorum, volume 10, pages 175–209. MR 2868943
2. ^ B. Gibert (2000): [ Message 1270]. Entry in the Hyacinthos online forum, 2000-08-22. Accessed on 2014-10-09.
3. ^ Dao Thanh Oai (2014), A Simple Proof of Gibert’s Generalization of the Lester Circle Theorem Forum Geometricorum, volume 14, pages 201–202. MR 3208157
4. ^

## References

• Clark Kimberling, "Lester Circle", Mathematics Teacher, volume 89, number 26, 1996.
• June A. Lester, "Triangles III: Complex triangle functions", Aequationes Mathematicae, volume 53, pages 4–35, 1997.
• Michael Trott, "Applying GroebnerBasis to Three Problems in Geometry", Mathematica in Education and Research, volume 6, pages 15–28, 1997.
• Ron Shail, "A proof of Lester's Theorem", Mathematical Gazette, volume 85, pages 225–232, 2001.
• John Rigby, "A simple proof of Lester's theorem", Mathematical Gazette, volume 87, pages 444–452, 2003.
• J.A. Scott, "On the Lester circle and the Archimedean triangle", Mathematical Gazette, volume 89, pages 498–500, 2005.
• Michael Duff, "A short projective proof of Lester's theorem", Mathematical Gazette, volume 89, pages 505–506, 2005.
• Stan Dolan, "Man versus Computer", Mathematical Gazette, volume 91, pages 469–480, 2007.